Algebra Examples

$(5 \sqrt{2} + 3 \sqrt{5}) (2 \sqrt{10} - 5)$

Step 1

Expand $(5 \sqrt{2} + 3 \sqrt{5}) (2 \sqrt{10} - 5)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$5 \sqrt{2} (2 \sqrt{10} - 5) + 3 \sqrt{5} (2 \sqrt{10} - 5)$

Step 1.2

Apply the distributive property.

$5 \sqrt{2} (2 \sqrt{10}) + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10} - 5)$

Step 1.3

Apply the distributive property.

$5 \sqrt{2} (2 \sqrt{10}) + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $5 \sqrt{2} (2 \sqrt{10})$ .

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Step 2.1.1.1

Multiply $2$ by $5$ .

$10 \sqrt{2} \sqrt{10} + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.1.2

Combine using the product rule for radicals.

$10 \sqrt{10 \cdot 2} + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.1.3

Multiply $10$ by $2$ .

$10 \sqrt{20} + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.2

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.1.2.1

Factor $4$ out of $20$ .

$10 \sqrt{4 (5)} + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.2.2

Rewrite $4$ as $2^{2}$ .

$10 \sqrt{2^{2} \cdot 5} + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.3

Pull terms out from under the radical.

$10 (2 \sqrt{5}) + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.4

Multiply $2$ by $10$ .

$20 \sqrt{5} + 5 \sqrt{2} \cdot - 5 + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.5

Multiply $- 5$ by $5$ .

$20 \sqrt{5} - 25 \sqrt{2} + 3 \sqrt{5} (2 \sqrt{10}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.6

Multiply $3 \sqrt{5} (2 \sqrt{10})$ .

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Step 2.1.6.1

Multiply $2$ by $3$ .

$20 \sqrt{5} - 25 \sqrt{2} + 6 \sqrt{5} \sqrt{10} + 3 \sqrt{5} \cdot - 5$

Step 2.1.6.2

Combine using the product rule for radicals.

$20 \sqrt{5} - 25 \sqrt{2} + 6 \sqrt{10 \cdot 5} + 3 \sqrt{5} \cdot - 5$

Step 2.1.6.3

Multiply $10$ by $5$ .

$20 \sqrt{5} - 25 \sqrt{2} + 6 \sqrt{50} + 3 \sqrt{5} \cdot - 5$

Step 2.1.7

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 2.1.7.1

Factor $25$ out of $50$ .

$20 \sqrt{5} - 25 \sqrt{2} + 6 \sqrt{25 (2)} + 3 \sqrt{5} \cdot - 5$

Step 2.1.7.2

Rewrite $25$ as $5^{2}$ .

$20 \sqrt{5} - 25 \sqrt{2} + 6 \sqrt{5^{2} \cdot 2} + 3 \sqrt{5} \cdot - 5$

Step 2.1.8

Pull terms out from under the radical.

$20 \sqrt{5} - 25 \sqrt{2} + 6 (5 \sqrt{2}) + 3 \sqrt{5} \cdot - 5$

Step 2.1.9

Multiply $5$ by $6$ .

$20 \sqrt{5} - 25 \sqrt{2} + 30 \sqrt{2} + 3 \sqrt{5} \cdot - 5$

Step 2.1.10

Multiply $- 5$ by $3$ .

$20 \sqrt{5} - 25 \sqrt{2} + 30 \sqrt{2} - 15 \sqrt{5}$

Step 2.2

Subtract $15 \sqrt{5}$ from $20 \sqrt{5}$ .

$5 \sqrt{5} - 25 \sqrt{2} + 30 \sqrt{2}$

Step 2.3

Add $- 25 \sqrt{2}$ and $30 \sqrt{2}$ .

$5 \sqrt{5} + 5 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{5} + 5 \sqrt{2}$

Decimal Form:

$18.25140769 \dots$